From a loop population to the background spectrum

This chapter uses the two before it, What the tension of a cosmic string measures and How a cosmic-string loop radiates and shrinks. You need the Dirac delta function, a change of variables in an integral and the radiation-era relations $a\propto t^{1/2}$ and $H=1/(2t)$. Units have $c=1$; times are measured in units of $1/H_0$ when numbers appear.

Counting loops of each length

Write the number of loops born per unit volume and time as $R(t_{\mathrm i})$. In Model I every loop is born with length $\alpha t_{\mathrm i}$, and the rate is

Here $\tilde c=0.23$ is the chopping efficiency, $\bar v_{\mathrm r}=0.662$ and $\xi_{\mathrm r}=0.271$ describe the long strings, $f_{\mathrm r}=\sqrt2$ removes the energy that goes into the loops' bulk motion, and $F=0.1$ accounts for the spread of real birth sizes. Auclair et al., 2020, Section 3.1, eqs. 3.12 to 3.16

A loop of length $\ell$ seen at time $t$ was born at the time that satisfies $\alpha t_{\mathrm i}-\Gamma G\mu\,(t-t_{\mathrm i})=\ell$:

Loops born in an interval $\mathrm dt_{\mathrm i}$ occupy a length interval $\mathrm d\ell=(\alpha+\Gamma G\mu)\,\mathrm dt_{\mathrm i}$ today, and their density has been diluted by $(a_{\mathrm i}/a)^3$. So

In the radiation era $(a_{\mathrm i}/a)^3=(t_{\mathrm i}/t)^{3/2}$. Substituting $t_{\mathrm i}$ and writing $x=\ell/t$ gives

which is eq. 3.16 of Auclair and colleagues. NANOGrav writes the same density for any era with the extra term $\dot\alpha_\ast t_\ast$ in the Jacobian, needed when the birth size changes with time. NANOGrav Collaboration, 2023, Section 5.4, eqs. 43 and 44

Adding up the emission

A loop of length $\ell$ at time $t$ emits power $G\mu^2P_n$ at frequency $2n/\ell$, which arrives today at $f=2n/[(1+z)\ell]$. Energy in gravitational waves dilutes as $(1+z)^{-4}$, and a frequency interval is stretched by $(1+z)$. The energy density per unit frequency today is therefore

The delta function picks $\ell_n=2n/[(1+z)f]$ with weight $\ell_n^2/(2n)$. With $\mathrm dt=\mathrm dz/[(1+z)H]$ and $\Omega_{\mathrm{gw}}=(8\pi G/3H_0^2)\,f\,\mathrm d\rho_{\mathrm{gw}}/\mathrm df$,

eqs. 4.4 and 4.5 of Auclair and colleagues. Auclair et al., 2020, Section 4.1 Because $C_n(f)=C_1(f/n)/n$, one table of the fundamental $C_1$ gives every harmonic.

The radiation-era plateau

At high frequency the integral is dominated by loops from the radiation era, where $H=H_0\sqrt{\Omega_{\mathrm r}}\,(1+z)^2$ and $t=1/(2H)$. Change variables from $z$ to $x=\ell_n/t$:

Every power of $(1+z)$ cancels, and $C_n(f)=(8H_0^2\Omega_{\mathrm r}/f)\int\mathsf n_{\mathrm r}(x)\,\mathrm dx$. The factor $f$ cancels too: the spectrum is flat. Summing $\sum_nP_n=\Gamma$ and integrating $\mathsf n_{\mathrm r}$ from $0$ to $\alpha$,

This is eq. 6.1. For large loops, $\epsilon\gg1$, it becomes $(128\pi/9)A_{\mathrm r}\Omega_{\mathrm r}h^2\sqrt{G\mu\,\alpha/\Gamma}$. Auclair et al., 2020, Section 6.2.1

Where the plateau formula stops

The plateau assumes radiation domination for the whole life of the loops that contribute. At low frequency the signal comes from large loops that radiate after matter takes over, the spectrum rises into a bump, and the formula fails. For $G\mu=10^{-10}$ the numerical fundamental-mode spectrum at one cycle per year is $2.43$ times the plateau formula. The Lab blends the radiation and matter loop densities by the radiation share of the energy density, a teaching choice; Auclair and colleagues instead solve the one-scale model through the transition.

Lab task

In the cosmic-string Lab, drag the loop size from its default down to the smallest value while watching the dashed plateau line. The line follows the curve at high frequency for every size. Then choose Loops born tiny and compare the plateau with the small-loop limit in exercise 3.

Exercises

1. Derive the matter-era density $\mathsf n_{\mathrm m}$ from the same steps, using $a\propto t^{2/3}$.
2. By what factor does the large-loop plateau change between $G\mu=10^{-10}$ and $G\mu=10^{-16}$ at fixed $\alpha$?
3. Show that for $\epsilon\ll1$ the plateau no longer depends on $\alpha$, and evaluate it for $G\mu=10^{-10}$.
4. The Lab's tiny-loop preset has $\alpha=10^{-8}$ and $G\mu=10^{-10}$. Evaluate eq. 6.1 and compare with the Lab's $25\,\mathrm{Hz}$ readout of $2.09\times10^{-13}$.